L10a87

L10a86

L10a88

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Other depiction

Mongolian ornament

Link Presentations

[edit Notes on L10a87's Link Presentations]

Planar diagram presentation	X8192 X20,9,7,10 X6718 X2,11,3,12 X12,3,13,4 X16,5,17,6 X18,13,19,14 X14,17,15,18 X4,15,5,16 X10,19,11,20
Gauss code	{1, -4, 5, -9, 6, -3}, {3, -1, 2, -10, 4, -5, 7, -8, 9, -6, 8, -7, 10, -2}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ -\frac{4 u^2 v^2-4 u^2 v+u^2-4 u v^2+9 u v-4 u+v^2-4 v+4}{u v} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{6}{q^{9/2}}+\frac{2}{q^{7/2}}-\frac{1}{q^{5/2}}-\frac{1}{q^{25/2}}+\frac{3}{q^{23/2}}-\frac{6}{q^{21/2}}+\frac{9}{q^{19/2}}-\frac{11}{q^{17/2}}+\frac{12}{q^{15/2}}-\frac{11}{q^{13/2}}+\frac{8}{q^{11/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^3 a^{11}+z a^{11}-z^5 a^9-z^3 a^9+z a^9-2 z^5 a^7-5 z^3 a^7-2 z a^7+a^7 z^{-1} -z^5 a^5-3 z^3 a^5-3 z a^5-a^5 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^5 a^{15}+2 z^3 a^{15}-z a^{15}-3 z^6 a^{14}+6 z^4 a^{14}-3 z^2 a^{14}-4 z^7 a^{13}+6 z^5 a^{13}-z^3 a^{13}-z a^{13}-3 z^8 a^{12}+z^6 a^{12}+3 z^4 a^{12}-z^9 a^{11}-5 z^7 a^{11}+6 z^5 a^{11}+3 z^3 a^{11}-3 z a^{11}-5 z^8 a^{10}+3 z^6 a^{10}+4 z^4 a^{10}-3 z^2 a^{10}-z^9 a^9-4 z^7 a^9+4 z^5 a^9+2 z^3 a^9-z a^9-2 z^8 a^8-3 z^6 a^8+10 z^4 a^8-6 z^2 a^8-3 z^7 a^7+4 z^5 a^7-z^3 a^7-z a^7+a^7 z^{-1} -2 z^6 a^6+3 z^4 a^6-a^6-z^5 a^5+3 z^3 a^5-3 z a^5+a^5 z^{-1} }[/math] (db)

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]).

\   r    \    j   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                     1 1 -6                   2 1 -1 -8                 4     4 -10               4 2     -2 -12             7 4       3 -14           5 4         -1 -16         6 7           -1 -18       4 6             2 -20     2 5               -3 -22   1 4                 3 -24   2                   -2 -26 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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